Let \(P\in \mathbb {F}_{n}(E)\)  be a central configuration, with mass-norm \(r={\Vert {P}\Vert }_M\) . As in the case \(r=1\) , seen in (CC1), it is a critical point of the restriction of the potential function \(U\)  to the inertia ellipsoid \(S= \lbrace  {{q}}\in \mathbb {F}_{n}(E) \colon {\Vert {{{q}}}\Vert }_M = r \rbrace \) . As such, its Morse index is the Hessian of the restriction \(U|_{S}\) , which is a bilinear form defined on the tangent space \(T_P S\) . The Hessian of \(f=U|_{S}\)  at a critical point \(P\in S\)  can be computed in general as \(D^2f(P)[{{u}}(P),{{v}}(P)] = ((D_{{u}}D_{{v}}f - D_{D_{{u}}{{v}}}) f) (P) = (D_{{u}}(D_{{v}}f))(P) \) , where \({{u}}\)  and \({{v}}\)  are vector fields on \(S\)  (see, e.g., [1]}). This yields the well-known formula of the Hessian in terms of second derivatives with respect to a local chart (see also [2]}). Given the mass-metric, the Hessian can be written as \(D^2f[{{u}},{{v}}] = {\langle { D_{{u}}\hat{\nabla }_Mf },{ {{v}}}\rangle }_{M}\)  or as the self-adjoint endomorphism \(T_PS \rightarrow T_PS\)  defined by \({{u}}\in T_PS \mapsto D_{{u}}\hat{\nabla }_Mf(P) \in T_PS\) , where \(\hat{\nabla }_M\)  denotes the gradient induced by the mass-metric restricted to \(S\)  (hence \(\hat{\nabla }_Mf (P)\)  is the projection of \(\nabla _MU(P)\)  to \(T_PS\) , orthogonal with respect to \({\langle {-},{-}\rangle }_{M}\) ).
